Theorem lautcnv 40595

Description: The converse of a lattice automorphism is a lattice automorphism. (Contributed by NM, 10-May-2013.)

Hypothesis

Ref	Expression
lautcnv.i	⊢ 𝐼 = (LAut‘𝐾)

Assertion

Ref	Expression
lautcnv	⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Proof of Theorem lautcnv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2741	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		lautcnv.i	. . . 4 ⊢ 𝐼 = (LAut‘𝐾)
3	1, 2	laut1o 40590	. . 3 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → 𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
4		f1ocnv 6782	. . 3 ⊢ (𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
5	3, 4	syl 17	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾))
6		eqid 2741	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
7	1, 6, 2	lautcnvle 40594	. . 3 ⊢ (((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
8	7	ralrimivva 3184	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))
9	1, 6, 2	islaut 40588	. . 3 ⊢ (𝐾 ∈ 𝑉 → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
10	9	adantr 482	. 2 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → (◡𝐹 ∈ 𝐼 ↔ (◡𝐹:(Base‘𝐾)–1-1-onto→(Base‘𝐾) ∧ ∀𝑥 ∈ (Base‘𝐾)∀𝑦 ∈ (Base‘𝐾)(𝑥(le‘𝐾)𝑦 ↔ (◡𝐹‘𝑥)(le‘𝐾)(◡𝐹‘𝑦)))))
11	5, 8, 10	mpbir2and 720	1 ⊢ ((𝐾 ∈ 𝑉 ∧ 𝐹 ∈ 𝐼) → ◡𝐹 ∈ 𝐼)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   class class class wbr 5074  ^◡ccnv 5619  –1-1-onto→wf1o 6487  ‘cfv 6488  Basecbs 17174  lecple 17222  LAutclaut 40490

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5201  ax-sep 5220  ax-nul 5230  ax-pow 5296  ax-pr 5364  ax-un 7681

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3725  df-csb 3833  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-pw 4533  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4841  df-iun 4925  df-br 5075  df-opab 5137  df-mpt 5156  df-id 5515  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7362  df-oprab 7363  df-mpo 7364  df-map 8769  df-laut 40494

This theorem is referenced by:  ldilcnv  40620